On root posets for noncrystallographic root systems

We discuss properties of root posets for finite crystallographic root systems, and show that these properties uniquely determine root posets for the noncrystallographic dihedral types and type $H_3$, while proving that there does not exist a poset satisfying all of the properties in type $H_4$. We do this by exhaustive computer searches for posets having these properties. We further give a realization of the poset of type $H_3$ as restricted roots of type $D_6$, and conjecture a Hilbert polynomial for the $q,t$-Catalan numbers for type $H_4$.